5 Pages
English

Jean Pierre Wintenberger

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Niveau: Supérieur, Doctorat, Bac+8
Jean-Pierre Wintenberger Extensions of Iwasawa modules (2) jw with Shekhar Khare The aim of the talk is show how Leopoldt conjecture is linked with prop- erties of exact sequences of Iwasawa modules arising from ramification at auxiliary primes. The hope is to be able to use modular technics to study these properties. We restrict to the case of a totally real F . Let p > 2 be a prime. Let F∞ = F (µp∞) be the cyclotomic extension. The cyclotomic character ?p identifies Gal(F∞/F ) to an open subgroup of (Zp)?, hence the quotient by its torsion is isomorphic to Zp. To this quotient, corresponds the cyclotomic Zp-extension we call F∞. A formulation of Leopoldt conjecture (LC) is that F∞ is the only Zp- extension of F . Let EF be the group of units of F . Let UF be the units in the p-adic completion of OF , so UF = ∏ ? U? where the ? are the primes of F over p. The group UF is the product the multiplicative groups ∏ ?(k?) ? of the residue fileds by the group U1F of units that reduces to 1 in ∏ ? k?. U 1 F is a Zp-module of rank r where r = [F : Q].

  • iwasawa

  • galois group over

  • let m∞

  • kummer extension

  • residue fields

  • adic modular

  • galois over

  • y∞ ?

  • ?i reduce


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Language English
Jean-Pierre Wintenberger
Extensions of Iwasawa modules (2)
jw with Shekhar Khare
The aim of the talk is show how Leopoldt conjecture is linked with prop-erties of exact sequences of Iwasawa modules arising from ramification at auxiliary primes.The hope is to be able to use modular technics to study these properties. We restrict to the case of a totally realF. Letp >Let2 be a prime. F=F(µpThe cyclotomic character) be the cyclotomic extension.χp identifies Gal(F/F) to an open subgroup of (Zp) , hence the quotient by its torsion is isomorphic toZp. Tothis quotient, corresponds the cyclotomic Zp-extension we callF. A formulation of Leopoldt conjecture (LC) is thatFis the onlyZp-extension ofF. LetEFbe the group of units ofF. LetUFbe the units in thep-adic Q completion ofOF, soUF=Uwhere theare the primes ofFover Q p. ThegroupUF(is the product the multiplicative groupsk) ofthe Q 1 1 residue fileds by the groupUof units that reduces to 1 ink.Uis a F ℘F Zp-module of rankrwherer= [F:Qhas an injection]. OneEFUF. Let ¯ EFbe the closure of the image ofEFinUF(with itsp-adic or congruence topology). LetLbe the maximal abelianp-extension ofFthat is only ramified atpfield theory gives an exact sequence :. Class
¯ 1UF/EFGal(L/F)Cl(F)1.
As aZp-extension is unramified outsidep, it follows that LC is equivalent ¯ to the fact that theZprank ofEFisr1,i.e.the rank ofEF. Itis equivalent to the fact that the topology onEFinduced by the topology of UFis the same as thep-adic topology.On can express this by the fact that ap-adic regulator made fromEFand thep-adic logarithms is non zero.By thep-adic formula of Colmez for the residue of thep-adicL-functionζF,p(s) ats= 1, it is equivalent to the fact thatζp(s) has a pole ats= 1. RemarkLC is known for abelian extensions ofQand imaginary quadratic fields by Brumer using independence of logarithms technics and Galois prop-erties of units. One has another formulation by Iwasawa.Letqbe a primes ofFprime top. IfLis a finite abelianp-extension ofF, the inertia subroupIq(L/F) of Gal(L/F) is a quotient of (kq) killedby a power ofp, hence it is cyclic of order divisible by thep-parte(q) ofN(q)call1. IwasawaLfully ramified ifIq(L/F) has ordere(qproved that the Leopoldt conjecture). Iwasawa is true forFandpif and only if , for eachqprime ofFprime top,F 1